It is known that a lattice is representable as a ring of sets iff the lattice is distributive.
CRL is the class of bounded distributive lattices (DLs) which have representations preserving arbitrary joins and meets.
jCRL is the class of DLs which have representations preserving arbitrary joins,
mCRL is the class of DLs which have representations preserving arbitrary meets, and
biCRL is defined to be
\({{\bf jCRL} \cap {\bf mCRL}}\) . We prove
${\bf CRL} \subset {\bf biCRL} = {\bf mCRL} \cap {\bf jCRL} \subset {\bf mCRL} \neq {\bf jCRL} \subset {\bf DL}$
where the marked inclusions are proper.
Let